Problem: $g'(x)=5g(x)$, and $g(3)=2$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $g(x)=e^{5x-6}$ (Choice B) B $g(x)=3e^{5x-10}$ (Choice C) C $g(x)=2e^{5x-15}$ (Choice D) D $g(x)=2e^{5x}$
Solution: The general solution of equations of the form $g'(x)=kg(x)$ is $g(x)=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=5$, so $g(x)=C\cdot e^{5x}$. Let's use the fact that $g(3)=2$ to find $C$ : $\begin{aligned} g(x)&=C\cdot e^{5x} \\\\ g(3)&=C\cdot e^{5\cdot 3} \gray{\text{Plug }x=3} \\\\ 2&=C\cdot e^{5\cdot 3} \gray{g(3)=2} \\\\ 2e^{-15}&=C \end{aligned}$ In conclusion, $g(x)=2e^{5x-15}$.